MATHEMATICS
PROTOTYPE
MATHEMATICS
LEARNER’S BOOK
SENIOR ONE
PROTOTYPE
LOWER SECONDARY
CURRICULUM
PROTOTYPE
PROTOTYPE
MATHEMATICS
LEARNER’S BOOK
SENIOR ONE
LOWER SECONDARY
CURRICULUM
DISCLAIMER!!
This material has been developed strictly for training purposes. Content and images
have been adapted from several sources which we might not fully acknowledge. This
document is therefore restricted from being reproduced for any commercial purposes
National Curriculum Development Centre
P.O. Box 7002,
Kampala- Uganda
www.ncdc.co.ug
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MATHEMATICS
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Contents
Acknowledgements ........................................................................................... viii
Topic 1 .................................................................................................................. 1
NUMBER BASES .................................................................................................... 1
Sub-topic 1. 1 Identifying numbers of different bases on an abacus .............. 2
Sub-topic 1. 2: Place Values Using the Abacus ............................................... 4
1.3 Converting Numbers ................................................................................ 5
1.4: Operation on Numbers in Various Bases .................................................. 6
Topic 2: ............................................................................................................... 11
WORKING WITH INTEGERS ................................................................................ 11
Introduction ................................................................................................... 12
Subtopic 2.1 Natural Numbers ...................................................................... 12
Sub topic 2.2: Differentiate between natural numbers and whole
numbers/integers. ......................................................................................... 14
Sub-topic 2.3: Use Directed Numbers (Limited to Integers) in Real-Life
Situations ....................................................................................................... 14
Sub-topic 2.4: Use the Hierarchy of Operations to Carry Out the Four
Mathematical Operations on Integers ........................................................... 17
Sub-topic 2.5: Identify Even, Odd, Prime and Composite Numbers .............. 24
Sub-topic 2.6: Find the Prime Factors of any Number ................................... 25
Sub-topic 2.7: Relate Common Factors with HCF and Multiples with LCM ... 27
Sub-topic 2.8: Work Out and Use Divisibility Tests of Some Numbers .......... 27
Sub-topic 2.9: Least Common Multiple (LCM) ............................................... 28
Topic 3: ............................................................................................................... 30
FRACTIONS, PERCENTAGES AND DECIMALS ...................................................... 30
Sub-topic 3.1: Describe Different Types of Fractions .................................... 31
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Sub-topic 3.2: Convert Improper Fractions to Mixed Numbers and Vice
Versa .............................................................................................................. 32
Sub-topic 3.3: Operations on Fractions . ....................................................... 33
3.33 Subtraction of Fractions with Same Denominators ............................... 37
3.34 Subtraction of Fractions with Different Denominators ......................... 38
3.35 Addition of Mixed Fractions .................................................................. 39
3.36 Subtraction of Mixed Fractions .............................................................. 40
3.37 Subtraction of Fractions with Different Denominators ......................... 44
3.38 Multiplication of Fractions ..................................................................... 45
3.39 Multiplying Mixed Fractions .................................................................. 47
3.310 Division of Mixed Fractions Flip And Multiply ..................................... 49
Sub-topic 3.5: Convert Fractions to Decimals and Vice Versa ....................... 52
Sub-topic 3.6: Identify and Classify Decimals as Terminating, Nonterminating
and Recurring Decimals ............................................................. 52
Sub-topic 3.7: Convert Recurring Decimals into Fractions ............................ 53
Sub-topic 3.8: Convert Fractions and Decimals into Percentages and Vice
Versa .............................................................................................................. 54
Sub –topic 3.9 Calculate a Percentage of a Given Quantity .......................... 55
Sub-topic 3.10: Works out Real-life Problems Involving Percentages ........... 56
Topic 4: ............................................................................................................... 58
RECTANGULAR CARTESIAN COORDINATES IN 2 DIMENSIONS .......................... 58
Sub-topic 4.2: Plotting Polygons (shapes) ...................................................... 60
Topic 5: ............................................................................................................... 64
GEOMETRIC CONSTRUCTION SKILLS .................................................................. 64
Sub-topic 5.2: Construction of Perpendicular lines ...................................... 66
Sub –topic 5.2: Using a Ruler, Pencil and Pair of Compasses, Construct
Special Angles ................................................................................................ 67
Sub-topic 5.3: Describing Locus Question...................................................... 68
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5.31: Relating Lines and Angles to Loci .......................................................... 68
Sub-topic 5.4: Construction of Geometric Figure .......................................... 69
Topic 6: ............................................................................................................... 71
SEQUENCE AND PATTERNS ................................................................................ 71
Sub-topic 6.2: Describing the general rule ..................................................... 79
Sub-topic 6.3: Generating Number Sequence ............................................... 80
Sub-topic 6.4 : Formulae for General Terms .................................................. 83
Topic 7: ............................................................................................................... 86
BEARINGS ........................................................................................................... 86
Sub-topic 7.1: Angles and Turns .................................................................... 87
Sub-topic 7.2: Bearings .................................................................................. 89
Topic 8: ............................................................................................................... 91
GENERAL AND ANGLE PROPERTIES OF GEOMETRIC FIGURES ........................... 91
8.1: Identify Different Angles ........................................................................ 93
Sub- topic 8.2 : Angles on a Line and Angles at a Point ................................. 94
Topic 9: ............................................................................................................... 97
DATA COLLECTION AND PRESENTATION ........................................................... 97
Sub-topic 9.1: Types of Data .......................................................................... 97
Sub-topic 9.2: Collecting Data ........................................................................ 99
Topic 10: ........................................................................................................... 105
REFLECTION ...................................................................................................... 105
Sub-topic 10.2: Reflection in the Cartesian Plane........................................ 107
Topic 11: Equation of Lines and Curves ........................................................... 108
Fundamental Algebraic Skills ....................................................................... 108
Subtopic 11.1 Function Machines ................................................................ 110
Sub-topic 11.2: Linear Equations ................................................................. 112
Topic 14: Time and Time Tables....................................................................... 116
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Sub-topic 14.1: Telling the Time .................................................................. 116
Sub-topic 14.2:12-hour and 24-hour Clocks ................................................ 118
Sub-topic 14.4: Timetables .......................................................................... 121
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Preface
This Learner’s Textbook has been written in line with the revised
Mathematics syllabus. The knowledge and skills which have been
incorporated are what is partly required to produce a learner who has
the competences that are required in the 21st century.
This h as b een d one b y p roviding a r ange o f a ctivities which will be
conducted both within and outside the classroom setting. The learner
is e xpected to be able to work a s an individual, in pairs and g roups
according to the nature of the activities.
The teacher as a facilitator will prepare what the learners are to learn
and this learner’s book is one of the materials which are to be used to
support the teaching and learning process.
Associate Professor Betty Ezati
Chairperson, NCDC Governing Council
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Acknowledgements
National Curriculum Development Centre (NCDC) would like to express
its appreciation to all those who worked tirelessly towards the
production of the Learner’s Textbook.
Our gratitude goes to the various institutions which provided staff who
worked as a panel, the Subject Specialist who initiated the work and
the Production Unit at NCDC which ensured that the work produced
meets the required standards. Our thanks go to Enabel which provided
technical support in textbook development.
The Centre is indebted to the learners and teachers who worked with
the NCDC Specialist and consultants from Cambridge Education and
Curriculum Foundation.
Last but not least, NCDC would l ike to acknowledge all those behind
the scenes who formed part of the team that worked hard to finalise
the work on this Learner’s Book.
NCDC is committed to uphold the ethics and values of publishing. In
developing this material, several sources have been referred to which
we might not fully acknowledge.
We welcome any suggestions for improvement to continue making our
service delivery better. Please get to us through P. O. Box 7002
Kampala or email us through admin@ncdc.go.ug.
Grace K. Baguma
Director, National Curriculum Development Centre
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Topic 1
NUMBER BASES
Key Words
base
binary
decimal
By the end of this topic, you should be able to:
i) identify numerals in base(s) up to base 16.
ii) identify place values of different bases using abacus.
iii) convert numbers from one base to another.
iv) manipulate numbers in different bases with respect to all four
operations.
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Introduction
I Am an ordinary person, how many fingers do I have on:
i) one hand?
ii) two hands?
If you have heaps of oranges of ten, twelve and fifteen, how many
groups of tens, fives and fours do you get in each? And how many are
remaining in each heap?
In order to answer the above questions, you can use your
knowledge of decimal place value to develop your understanding
of numbers written in other bases.
Sub-topic 1. 1: Identifying numbers of different
bases on an abacus
In your primary education, you studied number bases such as bases
five, two and ten (decimal base). Remember the numerals for all the
various number bases you studied by doing the following activity:
Activity 1. 1: Getting familiar with number bases
In your groups, identify situations in which you have ever used
number bases in your life.
Real life situation Base Reason for the base chosen
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Question: Which possible base does each abacus below represent?
a) b)
c)
d)
Activity 1.2: List the numerals for the following bases
In your groups, list the numerals for the following bases:
i) Two (Binary) ii) Three. iii) five iv) seven v) eight. vi) nine vii)
eleven viii) twelve ix) sixteen
Now study the table below and fill in the gaps.
Base Numerals
Two 0, 1
Three 0, 1, 2
Four 0, 1, -, 3
Five 0, - , 2, - , 4
Nine 0, 1, 2, - , 4, - , 6, - , 8
Twelve 0, - , 2, - , 4, - , - , 7, - , 9, - , e
Sixteen 0,1,2,3,4,5,-,-,-,9,t,e,-,-,-,-
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Compare your answers and note what happens to the base number
when writing the numerals used in a particular base. Give reasons.
Sub-topic 1. 2: Place Values Using the Abacus
You have already learnt how to represent numbers on an abacus. The
representation of numbers on an abacus helps you to identify the
place value of digits in any base.
Activity 1.3: Making abaci
In groups work in pairs to make different abaci, in different bases.
Compare your work with other members of the group .
Activity 1.4: Reading and stating the value of digits in bases
In groups, represent the following numbers on an abacus:
a. 123four
b. 274ten
c. 1312five
Read and state what each digit in the numbers above represents on an
abacus using the stated bases.
Exercise
State the place value of each numeral in the following numbers:
a) 321four b) 354six c) 247eight
State the value of each numeral in the following numbers:
b) 567nine b) 381twelve c) 11010two
Represent the following numbers on the abacus:
(a) 1101two (b) 2102three (c) 2021four (d) 5645seven (e) 8756nine
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1.3 Converting Numbers
Numbers can be converted from one base to another, and when you
do this, you get the same numbers written in different bases.
You learnt how to convert from base ten to any other base.
Activity 1.5: Converting numbers from base ten to any other base
In groups, convert the following numbers in base ten to bases
indicated: 456, 1321, 5693, 56 and 436.
(a) Five (b) Nine (c) Eight
You can also convert numbers from any base to base ten (decimal).
Example: Convert (a) 101two (b) 324five (c) 756eight to base ten.
Solution:
(a) 101two = (1x 22) + (0x21) + (1 x 20) = 1x4 + 0x1 + 1x1= 4+0+1 = 5
(b) 324five = (3x52) + (2x51) + (4x50) = 3x25 + 2x5 + 4x1 = 75+10+4 = 89
(c) 756eight= (7x82) +(5x81) +(6x80) = 7x64 + 5x8 + 6x1 = 448+40+6 = 494
Activity 1.6: Converting numbers in a given base to another base
In pairs, discuss how to convert numbers in different bases to various
bases in the exercise below.
Exercise
Convert the following numbers to the bases indicated: (a) 762eight to
base seven; (b) 234five to base six; (c) 561seven to base nine; (d) 654six to
base four; (e) 5432six to twelve.
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1.4: Operation on Numbers in Various Bases
James had two jackfruit trees in his compound. At one time one tree
had 8 fruits ready and the other 7 fruits. He harvested them at the
same time. He decided to put them in heaps of nine fruits. How many
heaps of nine did he get and how many remained?
When you put the fruits in heaps of 9, you are adding in base 9.
Addition
The two jack fruit trees above had a total of 15 (that is 8 +7) ready
fruits.
You can add numbers in various bases. For example, add the following
numbers:
(a) 234five to 23five (b) 153seven to 453seven
Solution
(a) (b)
Exercise: Add the following numbers:
(a)321four to 122four. (b) 456seven to 342seven
(c) 764eight to 361eight. (d) 210three to 211three
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Subtraction
Subtraction in other bases is done in the same way it is done in base
ten.
Examples: Subtract (a) 342eight from 567eight (b) 432six from 514six
Solution:
(a) (b)
Exercise
Subtract the following numbers in the given bases:
(a) 351six from 510six(b) 672nine from 854nine
(c) 845twelve from t23twelve(d) 231five from 421five
Multiplication
Multiplication is done in the way it is done in base ten.
Example: Multiply 423five by 12five
Solution
Exercise:
Multiply the following:
(a) 241five by 13five. (b) 345six by 24six
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(c) 534seven by 123seven. (c) 156eleven by 534eleven
Division
The most common method of dividing numbers in different bases is by
converting the numbers to base ten first and after division, you can
convert the answer to the given base.
Example: Divide 1111two by 101two
Solution: Convert 1111two and 101two to base ten
1111two = (1x23) + (1x22) + (1x21) + (1x20)
= 8 + 4 + 2 + 1
= 15.
101two = (1x22) + (0x21) + (1x20)
= 4 + 1 = 5ten
Therefore, 1111two divided 101two is the same as 15 divided 5.
15÷5 = 3
3ten = 3÷2 = 1 remainder 1 = 11two
Therefore, 1111two÷ 101two = 11two
Exercise:
1. Add: (a) 654seven to 514seven (b) 278nine to 756nine
2. Subtract: (a) 412six from 554six (b) 435eight from 764eight
3. Multiply: (a) 1121three by 212three (b) 312four by 122four
4. Divide: (a) 100011two by 111two (b) 150nine by 20nine
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Activity 1.6: Operations on numbers with mixed bases
In your groups work in pairs discuss how you would carry out the four
mathematical operations on numbers with mixed bases by getting
your own examples. Compare your answers with other members of the
group.
Number Game: You are given four boxes containing numbers in base
ten. The boxes are labelled Box 1, Box 2, Box 3 and Box 4.
9 1 15 7
Box 1
6 14 2 7 15
Box 2
15 14 6 12 4 7
Box 3
15 14 9 12
Box 4
Task: Working in groups, select one number from any of the boxes
given. Your mathematics teacher will ask you whether the number you
selected appears in Box 1, Box 2, Box 3 and Box 4. From the responses
you give, the teacher will tell you the number you selected. Discuss
how the teacher was able to tell you the number you had selected.
Situation of Integration
A community is hit by famine and the government decides to give each
member in the household a potato to solve their problem of hunger.
Support: Each package contains an equal number of potatoes of five.
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There are 10 households in the community with 3, 5, 7, 4, 6, 5,8,12, 13
members respectively.
Resources: Knowledge of Bases, knowledge of mathematical
operations
Task: Determine the number of packages of potatoes the government
will take to that community. In case there are remaining potatoes,
discuss what the government should do with them.
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Topic 2:
WORKING WITH INTEGERS
Key Words
positive, negative, BODMAS, LCM, HCF
By the end of this topic, you should be able to:
i) identify, read and write natural numbers as numerals and
words in million, billion and trillion.
ii) differentiate between natural numbers and whole
numbers/integers.
iii) identify directed numbers.
iv) use directed numbers (limited to integers) in real life situations.
v) use the hierarchy of operations to carry out the four
mathematical operations on integers.
vi) identify even, odd, prime and composite numbers.
vii) find the prime factorisation of any number.
viii) relate common factors with HCF and multiples with LCM.
ix) work out and use divisibility tests of some numbers.
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Introduction
Sarah was sent to a shop up the hill to buy 1kg of sugar, a packet of salt
and a packet of tealeaves. She was given UGX. 5,000 note but all items
cost her UGX. 6,500. How much money did Sarah owe the shopkeeper?
In your day-to-day life, you use numbers to count items, to keep
information, to transact business and many others. Since you use
numbers in your day-to-today situations, knowledge of integers will be
helpful to you.
Subtopic 2.1: Natural Numbers
In lower primary, you learnt counting items using numbers one, two,
three ---. In mathematics these numbers are called counting or natural
numbers.
When zero is included in the set of natural numbers, they become
whole numbers.
For example: N = {1,2,3,4,5,− − − − − −}This is a set of natural numbers.
W ={0,1,2,3,4,5. − − − − − −}This is a set of whole numbers.
Activity 2.1: Natural numbers
There is a box and a board. In the box, there are number cards: some
have numbers in figures and others in words. While the board has two
sections: one section for natural and the other for non-natural
numbers.
In groups, pick a card and place it in the appropriate section of the
board.
Is it possible for a number to belong to two sections?
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What can you say about the two categories of the numbers picked?
Where in real-life situations do we find such numbers?
Activity: 2.2: Writing and reading numbers
There are two boxes. In one box, number cards are written in figures
and the others in words.
In groups, a member picks one card from one of the boxes. After all the
cards have been picked, one member displays his/her card; then the
others check their cards, and the matching card is displayed.
Exercise
Write the following in words:
1. 3,800
2. 8,008,008
3. 606,520,060
4. 9,000,909,800
5. 4,629,842,003
6. 1,629,284,729,000
Write the following in figures:
7. Six hundred two million eight thousand and eight
8. Two billion eighty-nine million four thousand seven
9. One trillion two hundred fifty billion eight hundred seventy-five
million three hundred sixty thousand
10. State the value of digit four in the following numbers.
i) 7,462,300,800
ii) 24,629,293,005
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Sub topic 2.2: Differentiating between natural
numbers and whole numbers/integers
Activity 2.3: Relating natural numbers and integers
In groups, read the text below and answer the questions that follow:
Two learners—Mary and Joy—went to the school canteen to buy some
snacks for their breakfast. Joy bought 3 pancakes at UGX.200 each and
1 ban at UGX. 300.
Mary checked her bag and found out that her money was stolen. She
borrowed some money from Joy. She bought four 4 pancakes and 2
bans.
Questions
i) Which of the two learners had more money?
ii) How much money did Mary borrow from Joy?
iii) Peter said that Mary had negative UGX. 1400. Was he correct?
Give reasons for your answer.
Sub-topic 2.3: Use Directed Numbers (Limited to
Integers) in Real-life Situations
Activity 2.4: Integers in real-life situations
In groups, read the story below and answer the questions.
Once upon a time, there lived an old woman. She had hot and cold
stones and a big pot of water. If she put one hot stone in the water, the
temperature of the water would rise by 1 degree. If she took the hot
stone out of the water again, the temperature would go down by 1
degree.
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Question 1
If the temperature of the water is 24 degrees and the old woman adds
5 hot stones, what is the new temperature of the water?
Now imagine that the temperature of the water is at 29 degrees. The
old woman takes a spoon and takes out 3 of the hot stones from the
pot.
Question 2
What is the temperature of the water when the old woman removes 3
hot stones? Explain your answer.
The old woman also had cold stones. If she adds 1 cold stone to the
water, the temperature goes down by 1 degree. The temperature of
the water was 26 degrees. Then the old woman added 4 cold stones.
Question 3
What is the temperature of the water after the old woman added 4 cold
stones? Give a reason for your answer.
Just like the old woman could r emove t he h ot s tones a nd t he
temperature would decrease she could also remove the cold stones.
Question 4
Imagine that the temperature of the water was 22 degrees and the old
woman removes 3 cold stones. What happens to the temperature of
the water?
What is the new temperature of the water? Explain your answer.
Activity 2.5: Real-life situations
In groups, get a cup of hot water and a thermometer. Identify a
timekeeper in your group. One of you reads the temperature on the
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thermometer and the other members record in their notebooks. Put
the thermometer back into the hot water and after 5 minutes take the
reading on the thermometer. Repeat this at same interval of 5 minutes
for duration of 25 minutes.
Question 1
What is the change in temperature between the first reading and the
second reading?
Question 2
What is the change in temperature between the 2nd and 3rd reading?
Question 3
What is the change in temperature between the 3rd and the 4th
reading?
Question 4
What is the difference in temperature between the 4th and the 5th
reading?
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Sub-topic 2.4: Use the Hierarchy of Operations to
Carry out the Four Mathematical Operations on
Integers
Activity 2.6: Operations on integers
In groups, read the text below and answer the questions after.
Sarah moved 5 steps to the right from a fixed point. Then she moved 9
steps to the left.
Question 1
How far is Sarah from the fixed point?
Question 2
Peter gave his answer as 4 steps to the left of the fixed point and John
as -4 (negative 4). Who is correct? Give reasons for your answer.
Example 1
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Example 2: A group of learners of Geography went for a tour to Kabale.
They found out that the temperature at one time was 130C. At around
mid-night the temperature was 100C. By how many degrees had the
temperature dropped?
Solution: 100C - 130C = - 30C
Example 2: Using a number line work out:
a) – 4 + +6
b) +5 + - 9
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c) -6 - 4 = -6 + - 4
-6 - 4 = -6 + -4 = -10
Exercise
1. Work out the following in degrees:
Note - x- = +, + x + = +
-x + = - , - ÷ - = +
- ÷ + = -
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2. Work out the following:
a) 8 + -6
b) 61 + + 7
c) 49 - - 30
d) 77 - + 50
e) -15 + + 20
f) -3 - - 13
25
5
x
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3. 2.Using a number line work out:
a) -2 ++ 3
b) +5 + - 6
c) – 8 - -5
4. A national park guide on o ne o f t he m ountains i n E ast A frica
recorded the temperature as 150C one day. At midnight the
temperature was -70C. By how many degrees had the temperature
fallen?
5. Write down the next 3 terms in the sequence - 9, -7, - 5, -3, - , - ; -
6. Look at the sequence of the numbers:
-1, 3,-9, +27, -, - , -
Alex said the next three terms are +9, -36 and -729. Is Alex correct? Give
reasons for your answer.
Multiplication and Division
Multiplication such as +4 × + 3 or -4 × + 3 are interpreted as repeated
addition of positive or negative numbers.
+4 × + 3 = + 4 + +4 + +4 = +12
-4 × +3 = -4 + -4 + -4 = -12
Negative
3 × 3 = 9
3 ×- 3 = -9
-3 × -3 =9
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Justification of the above is as follows:
3 × 3 = 9
3 × 2 = 6
3 × 1 = 3
3 ×0 = 0
3 × -1 = -3
3 × -2 = -6
3 × -3 = -9
Now reduce the first multiplier
3 × -3 =-9
2 × -3 = -6
1 × -3 =-3
0 × -3 =0
-1 × -3 = 3
-2 × -3 = 6
-3 × -3 = 9
The justification shows that any number multiplied by zero is zero; that
a positive number multiplied by a positive number is a positive; a
negative number multiplied by a positive number is a negative, and a
negative number multiplied by a negative is a positive.
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Multiplication and division have the same rules:
A negative number divided by a positive and a positive number divided
by a negative number is a negative, Also a negative number divided by
a negative is a positive.
Example
+ 4 × -3 = -12
-12 ÷ - 3 = +4
- 12 ÷ +4 = - 3
Note: Rules of integers
a) Positive number multiplied by a positive number is a positive.
b) Negative number multiplied by a positive number is a negative.
c) Negative number multiplied by a negative is a positive.
d) Negative number divided by a positive is a negative
e) Positive number divided by a negative is a negative.
f) Negative number divided by a negative is a positive.
Exercise
Work out
1. - 2 ×+ 4 × -3
2. -4 ×+2 × - 3
3. -3× -5 × +2
4. -12 ×-5 ÷ +6
5 -15 ÷ 5 × -4
6. -24 × + 4 ÷+2
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7. In a certain test a correct answer scores 3marks and an incorrect
answer, the child gets a penalty of two marks deducted. Joy
guessed all the answers. She got 6 correct and 4 wrong. Work out
her total marks.
8. Simplify +6 - +7 ÷ +4 + + 6 × +7
9. Work out 7 of 13 – (18 ÷ 6 +3) ÷ (9 × 3 -25)
10. 56 - (38 - 35 ÷5 +2)
11. 69 ÷ (6 + (3 × 8 -7))
12. 4 of (5 + 2) - 2 (3 + 7) ÷ 5
Sub-topic 2.5: Identify Even, Odd, Prime and
Composite Numbers
Natural numbers can be classified into various groups of numbers. In
your primary education, you learnt numbers such as even, odd, prime
and composite.
Activity 2.6: Identifying even, odd, prime and composite numbers
Each group is given a box containing number cards. In your groups
pick the card and read the number. Identify which group of numbers it
belongs to by filling the table below.
No Odd Prime Even Composite
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Question 1
Are there numbers that belong to more than one group?
Question 2
How do you identify that a number is:
a) odd
b) even
c) prime
d) composite
Sub-topic 2.6: Find the Prime Factors of any
Number
In your primary education you studied multiples and factors of
numbers. When two numbers are multiplied together, the product is
called multiple. The two numbers multiplied together are called
factors of the multiple.
Note: A multiple has two or more factors.
For example: The factors of 12 are (1 × 12), (2 × 6 ) , and ( 3 × 4 );
hence, the factors of 12 are {1,2,3,4,6,12}= F12 = {1,2,3,4,6,12}
The multiples of 3 are {3,6,9,12,15,18,21− − −} = M3 =
{3,6,9,12,15,18,21− −}
Exercise
Find the factors of the following:
1. 42
2. 56
3. 36
4. 108
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Find the multiples of the following:
5. 7
6. 12
7. 9
8. 5
Note: A factor of a number which is a prime number is called its prime
factor. For example the factors of 36 are {1,2,3,4,6,9,12,36}
Qn. What are the prime factors of 36?
Qn. Write 36 as a product of its prime factors.
Answer:
Prime Factor Number
2 36
2 18
3 9
3 3
1
36 = 2× 2 × 3 × 3 = 22 × 32
This approach of determining prime factors is called prime
factorisation.
This can be written in power notation.
Exercise
Find the prime factors of the following numbers. Give your answer in
power form.
1. 108
2. 288
3. 180
4. 1232
5. 993
6. 2145
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Sub-topic 2.7: Relate Common Factors with HCF
and Multiples with LCM
A number can have one or more common factors; for example, 2 and 4
are common factors of 8 and12. However, the highest common factor
is 4. Therefore, the highest common factor (HCF) of 8 and 12 is 4.
Activity 2.7: Highest common factor
In groups, find the HCF of the following:
i) 54, 48
ii) 42 ,63 ,105
iii) 132, 156,204,228
Sub-topic 2.8: Work Out and Use Divisibility Tests
of Some Numbers
Activity 2.8: Identifying divisibility tests for some numbers
1. In your groups, pick a number card and determine which
numbers on the chart divides it. Write a number under its
divisor.
2. What can you say about the numbers under each divisor? Give
reasons for your answers.
3. The relationship between the dividend and the divisor leads to
divisibility tests.
Exercise
Given the following numbers:
12, 132, 1212, 3243, 1112, 81, 18, 27, 279, 2580, 5750
Find out which of them are divisible by:
a) 3 b) 4 c) 6 d) 9 e) 10
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Exercise
Find the HCF the following:
1. 2 × 2× 3 × 3× 3 ×3 × 5 × 5 ×5 × 11
2. 2 × 2 ×2 ×2 ×2 ×3 ×3 ×5 ×7 ×13
3. 2. 23 ×32×5 2, 25 × 35 ×52
4. 36, 60, 84
4. A rectangular field measures 616m by 456m. Fencing posts are
placed along its sides at equal distances. What will be the distance
between the posts if they are placed as far apart as possible? How
many posts are required?
Sub-topic 2.9: Least Common Multiple (LCM)
In the previous section of multiples and factors you learnt about
multiples of numbers. For example, the multiples of 5 are 5, 10,
15,20,25,30, 35, 40, 45, 50, 55, 60, 65, 70, 75 ------. The multiples of 7 are
7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77 ------. From the above example, 35
and 70 are common multiples of 5 and at the same time of 7. However,
35 is smaller than 70, therefore, 35 is the least common multiple of 5
and 7.
There is another approach of getting LCM of numbers without listing
the multiples of the numbers.
Example
Find the LCM of 8 and 12
2 8 12
2 4 6
2 2 3
3 1 3
1 1
2 × 2 × 2 × 3 = 24
The LCM of 8 and 12 is 24.
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Activity 2.9: In your groups, find the LCM of the following:
a) 28, 42 ,98
b) 35,48 ,56, 70
Exercise
Find the LCM of the following numbers:
1. 14, 21
2. 18, 24, 96
3. 49, 84, 63
4. 60, 72, 84, 112
5. Determine the smallest sum of money out of which a number of
men, women and children may receive UGX. 75, Ush.90 and
Ush.120 each.
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Topic 3:
FRACTIONS, PERCENTAGES AND
DECIMALS
Key Words
recurring, numerator, denominator, terminating,
non-terminating, reciprocal, whole
By the end of this topic, you should be able to:
i) describe different types of fractions.
ii) convert improper fractions to mixed numbers and vice versa.
iii) work out problems from real-life situations.
iv) add, subtract, divide and multiply decimals.
v) convert fractions to decimals and vice versa.
vi) identify and classify decimals as terminating, non-terminating
and recurring decimals.
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vii) convert recurring decimals into fractions.
viii) convert fractions and decimals into percentages and vice versa.
ix) calculate a percentage of a given quantity.
x) work out real-life problems involving percentages.
Introduction
In Chapter Two you studied place values in number bases. In this topic,
you will use knowledge of place values to manipulate fractions,
decimals and percentages. You will convert fractions to decimals,
decimals to percentages and vice versa.
Sub-topic 3.1: Describe Different Types of
Fractions
Activity 3.1
Create a park of different cards and label them with different types of
fractions, decimals and percentages.
From the park of the cards, you pick a card and place it in the most
appropriate play area.
Observe the fractions in each play area by looking at the denominators
and numerators.
In your groups explore and explain the common of the classification
made in the different play areas.
Exercise
1. Sarah shades 3/7 of a shape. What fraction of the shape is left
unshaded?
2. A cake is divided into 12 equal parts. John eats 3/12 of the cake
and Kate eats another 1/12. What fraction of the cake is left?
3. A car park contains 20 spaces. There are 17 cars parked in the car
park.
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a) What fraction of the car park is full?
b) What fraction of the car park is empty?
4. Ali eats 3/10 of the sweets in a packet.
Tariq eats another 4/10 of the sweets.
a) What fraction of the sweets has been eaten?
b) What fraction of the sweets is left?
5.
a) Draw a square with its four lines of symmetry.
b) Shade 3/8 of the shape.
c) Shade another 2/8 of the shape.
d) What is the total fraction now shaded?
e) How much is left unshaded?
Sub-topic 3.2: Convert Improper Fractions to
Mixed Numbers and Vice Versa
Mixed Numbers and improper Fractions
So far you have worked with fractions of the form a/b where a < b, e.g.
¾, 2/7, 5/6 …
You also need to work with what are sometimes called improper
fractions, e.g. 5/4, 7/2, which are of the form a/b when a and b are
whole numbers and a > b.
Example
Convert 13/4 into an improper fraction.
Solution
13 ÷ 4 = 3 remainder 1
This is written as 3 ¼.
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Exercise
1. Draw diagrams to show these improper fractions:
(a) 7/2 (b) 8/3 (c) 18/5
Write each improper fraction as a mixed number.
2. Convert these mixed numbers to improper fractions.
(a) 1 3/5 (b) 7 1/3 (c) 3 4/5 (d) 6 1/9
3. Write these fractions in order of increasing size.
6 ½ , 18/5 , 3 ¼ , 5 1/3 , 17/3
4. In an office there are 2 ½ packets of paper. There are 500 sheets of
paper in each full packet. How many sheets of paper are there in the
office?
5. A young child is 44 months old. Find the age of the baby in years as
a mixed number in the simplest form.
Sub-topic 3.3: Operations on Fractions
In the previous sub-topic, you studied how to find equivalent fractions.
In this sub-topic you are going to use the knowledge of equivalent
fractions to add and subtract fractions.
3.3.1: Work out problems from real-life situations
Now we start to use fractions in a practical way.
Example
(a) Find 1/5 of UGX. 10000
(b) Find 4/5 of UGX. 100,000
You can, do this practically, but it is much easier to work out.
(a) 1/5 of 10000 = 1/5 x 10000 = 2000
(b) 4/5 of 100000 = 4/5 x 100000 = 400000/5 = 80,000
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Exercise
1. Find:
(a) ½ of 12 (b) 1/8 of 40 (c) ¼ of 32
2. Find:
(a) 2/9 of 18 (b) 7/9 of 45 (c) 7/8 of 56
3. In a test, there are 30 marks. Nasim gets 3/5 of the marks. How many
marks does she get?
4. In a certain school there are 550 pupils. If 3/50 of the pupils are lefthanded,
how many left-handed pupils are there in the school?
Activity 3.3: Addition of Fractions
In your groups, use a sheet of paper to work out
5
3
5
1 + . Fold the paper
into five equal parts shade off one part of the five equal parts
Shade the three parts of the five equal parts
How many parts have been shaded?
Represent the shaded parts in a fraction form. Show the working.
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Activity 3.4: Addition of Fractions with the Same Denominators
Slice a hexagon into 6 pieces:
Each piece is of the hexagon. Right?
And is of the hexagon.
So, what if we wanted to add
Hmm... that would be
Count them up
So
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In your groups, use the same method to work out the following:
a)
7
2
7
3 +
b)
9
4
9
5 +
3.3.2: Adding Fractions with the different Denominators
In the previous topic you studied about lowest common multiple. In
this section, you will apply the knowledge of LCM.
Change the using the knowledge of equivalent fractions
Change the using the knowledge of equivalent fractions
The main rule of this game is that we cannot add the fractions until the
denominators are the same!
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We need to find something called the least common denominator
(LCD)..which is the LCM of our denominators, 2 and 3.
The LCM of 2 and 3 is 6. So, our LCD 6.
We need to make this our new denominator
3.3.3: Subtraction of Fractions with Same Denominators
Let's try
Look at a Chapatti in a conical shape cut into 8 pieces. Each piece
is of the Chapatti.
Take away (that's 3 pieces):
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We're left with 4 pieces - that’s.
So
But, look what we really did!
We just subtracted the numerators!
which is
3.3.4: Subtraction of Fractions with Different Denominators
Subtraction works the same way.
The LCM of 11 and 22 is 22... So, the LCD is 22.
We just need to change the .
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Done!
3.3.5: Addition of Mixed Fractions
What if we need to add
?
Hey, remember, that's just .
Done!
That was easy, but, what about mixed numbers?
How about this?
All we have to do is change these to improper fractions... Then we can
add them!
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3.3.6: Subtraction of Mixed Fractions
?
Well, we can't just stick it together like we would if it was addition.
We need to get a common denominator... But, the 5 does not even
have a denominator!
That's OK... Just think of a Chapatti cut into 8 pieces...
How many pieces would there be in 5 chapattis? Yep!
pieces
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So
Check it: is the same as which is . Yep!
Back to the problem:
What's ?
Well, that's of 6. Think about it:
You have 6 chapattis.
and you get to eat of them.
This is like splitting up the chapatti between 3 people:
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You get 2 chapattis
Your friend gets
2 chapattis
And your dog gets
2 chapattis
You get 2 chapattis
Your friend gets
2 chapattis
And your dog gets
2 chapattis
So of 6 is 2.
But, how do we do this with just math? EASY!
We know how to multiply two fractions... Right?
So, just make both things be fractions. Check it out:
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is already a fraction...
But, what about the 6?
Guess what? We can write 6 as .
Let's try
Look at a chapatti cut into 8 pieces. Each piece is of the
Chapatti.
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Take away (that's 3 pieces):
We're left with 4 pieces, that's.
So
But, look at what we really did!
We just subtracted the numerators!
which is
3.3.7: Subtraction of Fractions with Different Denominators
Subtraction works the same way.
The LCM of 11 and 22 is 22... So the LCD is 22.
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We just need to change the .
Done!
3.3.8: Multiplication of Fractions
What’s ?
Well, that's of 6. Think about it…
You have 6 chapattis.
and you eat of them.
This is like splitting up the pizza between 3 people:
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You get 2
chapattis
Your friend gets
2 chapattis
And your dog gets
2 chapattis
So of 6 is 2.
But, how do we do this with just math? EASY!
We know how to multiply two fractions... Right?
So, just make both things be fractions. Check it out:
is already a fraction...
But, what about the 6?
Guess what? We can write 6 as .
Think about it:
is the same as ... which is 6!
(You can do this with any number!)
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Back to the problem:
Just what we figured!
3.3.9: Multiplying Mixed Fractions
What about this?
Yikes! I am sure I don't want to try to think about pizza for this one!
Let's stick to the math:
Again, let's change these into improper fractions and go for it!
This is super easy!
Let's just do one:
We just multiply straight across...
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Now, think about it...
Cut a pizza into 10 pieces like
and look at 9 of the pieces:
We want of these
That would be 3 pieces. Right?
That's !
Doing math is cooooool!
Now that we understand what to do, we can just go for it.
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3.3.10: Division of Mixed Fractions Flip And Multiply
Check it out:
That's it -- then GO FOR IT!
Done!
Look at another one:
Use the same trick you do when multiplying by changing everything to
fractions and then go for it!
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Check it out:
How about another one?
Use the same trick you do when multiplying by changing everything
into fractions and then go for it!
Sub-topic 3.4: Add, Subtract, Divide and Multiply
Decimals
Activity 3.5: Fractions and decimals
In groups, copy and complete the table, by explaining how you have
obtained the answer. The first three have been done for you
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Tens Ones Tenth
(
10
1 )
Hundredth
(
100
1 )
Thousandth
(
1000
1 )
Fraction Percentage
5
2
1
50
1 2 4 12
5
2
1240
2 5
4
1
25
1 5 2
5
80
20
17
64
0 0 4
10
3
4 0 3
The column headings
will help you
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Sub-topic 3.5: Convert Fractions to Decimals
and Vice Versa
A fraction like ¾ means three quarters
or three parts out of four
or three divided by four
3 divided by 4 equals 0.75
So, the fraction ¾ is equal to 0.75 in decimal.
Activity 3.6: In pairs, convert the following fractions into decimals
a) 2/5
b) (b) 1/20 (c) 5/8 (d) 2/9 (e) 1/11
c) What do you notice about (d) and (e)?
Sub-topic 3.6: Identify and Classify Decimals as
Terminating, Non-terminating and Recurring
Decimals
Fractions like 3/5, 1/2, 3/8 can be converted into decimals and they end or
terminate: 3/5 = 0.6, ½ = 0.5 and 3/8 = 0.375.
Fractions like 2/3, 2/15, 1/11 do not end or terminate when converted
into decimals, 2/3 = 0.66666…, 2/15 = 0.133333… and
1/11 = 0.090909…
These decimals are referred to as recurring decimals
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Exercise
1. Write the following fractions as decimals:
(a) 3/8 (b) 7/10 (c) 17/50 (d) 13/25
2. Write the following as fractions in their lowest terms:
(a) 0.25 (b) 0.08 (c) 0.35 (d) 0.125
3. Write the following fractions as recurring decimals:
(a)a 2/11 (b) 1/3 (c) 1/6 (d) 7/9
Sub-topic 3.7: Convert Recurring Decimals into
Fractions
Recurring decimals can be converted into fractions.
Example: Convert this recurring decimal into a fraction: 0.333…
Note that the decimal repeats itself after one decimal place.
Let r = 0.333… (1)
Multiply both sides of the equation by 10 i.e. 10 x r = 10 x 0.333
10r = 3.333 (2)
Subtract equation (1) from equation (2):
That is, 10r = 3.333
- (r = 0.333)
9r = 3
r = 3/9 = 1/3.
Exercise
1. Convert the following recurring decimals into fractions
a) 0.77---, b) 0.133--- , c)1.25656 ---, d) 0.2727 ---, e) 0.01313
2. Convert the following numbers into recurring decimals
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a)
3
1
, b)
9
1 , c)
6
2
Sub-topic 3.8: Convert Fractions and Decimals
into Percentages and Vice Versa
Activity 3.7: Fraction percentage game
I am
20
7
Who is
67%?
I am
100
67
Who is
13%?
I am
100
13
Who is
22%?
I am
11
Who is
5%?
I am
1
Who
is72%?
I am
18
Who is
87%?
I am
87
Who is
4%?
I am
1
Who is
34%?
I am
8
Who is
42%?
I am
21
Who is
52%?
I am
13
Who is
45%?
I am
9
Who is
58%?
I am
29
Who is
64%?
I am
16
Who
is32%?
I am
17
Who
is2%?
I am
1
Who is
92%?
I am
23
Who is
98%?
I am
49
Who is
44%?
I am
11
Who is
82%?
I am
41
Who is
65%?
I am
13
Who is
14%?
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From the fraction percentage game, identify the equivalent
percentage for each fraction.
In your groups, use percentage to identify the smallest and largest
fractions from the fraction percentage game.
Sub –topic 3.9 Calculate a Percentage of a Given
Quantity
The percentage of a quantity can always be calculated in terms of
percentage increase or percentage decrease.
Example 1: Find the 10⁒ of 50,000
Solution: 10/100 x 50,000 = 5,000.
Example 2: Opio had 60 goats. Now he has 63 goats. What is the
percentage increase?
Solution: The increase in the number of goats is 63 – 60 = 3.
Percentage increase is 3/60 x 100 = 5⁒.
Activity 3.8: The table below shows students’ marks in two
mathematics tests. For each one, calculate the percentage
difference. Say if it is an increase or a decrease.
Student First Test Second Test
(a) Marion 50 45
(b) James 40 52
(c) Christina 20 35
(d) Sarah 60 50
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Sub-topic 3.10: Works out Real-life Problems
Involving Percentages
Exercise
1. In a closing-down sale, a shop offers 50% cut of the original
prices. What fraction is taken off the prices?
2. In a survey one in five people said they preferred a particular
brand of Coca Cola. What is this figure as a percentage?
3. Peter pays tax at the rate of 25% of his income. What fraction of
Peter’s income is this?
4. When Carol was buying a house, she had to make a deposit of
10
1
of the value of the house. What percentage was this?
5. I bought a coat in the January sales with
3
1 price cut of the selling
price. What percentage was taken off the price of the coat?
6. Adikinyi bought some fabric that was 1.75 metres long. How
could this be written as a fraction?
7. A car park contains 20 spaces. There are 17 cars parked in the car
park.
a. What fraction of the car park is full?
b. What fraction of the car park is empty?
Sub-topic 3.11: Identifying and classifying decimal
as terminating, non-terminating and recurring
decimals
Activity 3.6: Decimal as terminating, non-terminating and
recurring decimals
In groups list some terminating, none terminating and
recurring decimals. In pairs prove them. Compare your
answers with the members of the group.
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Situation of Integration
A primary school has two sections, that is, lower primary (P1-P4) and
upper primary (P5-P7). The head teacher of the primary school needs
to draw a timetable for both sections. The sections should start and
end their morning lessons at the same time before break time, start
and end their break time at the same time. The after break lessons
should start at the same time. The lunchtime for both sections should
start at the same time.
Support: The time to start lessons for the two sections is 8.00am. The
duration of the lesson for the lower section is 30 minutes and
that of the upper section is 40minutes.
Resources: Knowledge of fractions, percentages, natural numbers,
factors, multiples, lowest common multiples, and the
subjects taught in all classes and of time.
Task: Help the head teacher by drawing the timetable up to lunchtime
for the two sections. How many lessons does each section have
up to lunchtime?
Express the total number of lessons for the lower primary as a fraction
of the total number of lessons for the whole School. (Consider lessons
up to lunch time.)
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Topic 4:
RECTANGULAR CARTESIAN
COORDINATES IN 2 DIMENSIONS
Key words: coordinates, axes, plot, scale
By the end of this topic, you should be able to:
i) identify the y-axis and x-axis.
ii) draw and label the Cartesian plane.
iii) read and plot points on the Cartesian plane.
iv) choose and use appropriate scale for a given data set.
v) identify places on a map using coordinates (apply coordinates
in real-life situations).
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Introduction
This topic is key in building the concept of location. The knowledge
achieved from this topic can be used in locating places. In order to
locate places you need a starting point (reference point).
Sub-topic 4.1: Identify the X-axis and Y-axis
Activity 4.1: Plotting Points
Now, plot the following points on a graph, (6,4), (5,9), (11,3), (5,6) and
(3, 4).
The x number comes first then the y number: (X, Y). These numbers are
called coordinates.
Exercise
1. Use a graph paper to:
a) Join the points with coordinates (0, 3), (5,6), and (5,0) to draw a
triangle.
b) On the same diagram join the points with coordinates (2, 0), (2,
6) and (7, 3) to draw a second triangle.
c) Describe the shape you have now drawn.
2. On the same graph paper join these points in order.
a) (4, 6), (5, 7), (6, 6), (4, 6).
b) (5, 8), (4, 8), (4, 7), (5, 8), (6, 8), (6,7), (5, 8).
c) (4, 5), (5, 4), (6, 5), (5, 3), (4, 5).
d) (5, 2), (3, 4), (3, 5), (2, 5), (2, 8), (3, 8), (3, 9), (7, 9), (7, 8), (8, 8), (8,
5), (7, 5), (7, 4), (5, 2).
We can also use negative numbers in coordinates. We can bring in
coordinate axes with positive and negative numbers.
Exercise
1. (a) Draw a set of axes and mark the points with coordinates (4, 0), (-
4, 0), (0, 4),
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(0, -4), (1, 2), (1, -2), (3, 3), (3, -3), (2, 1), (2, -1), (-1, 2), (-1, -2), (-3, 3), (-3, -
3),(-2, 1), (-2, -1)
(b) Join the points to form an 8 pointed star.
2. (a) On a graph paper, draw the rectangles with corners at the
following points with coordinates:
a) (-6, 6), (-5, 6), (-5, 4), (-6, 4)
b) (-2, 1), (-3, 1), (-3, 3), (-2, 3)
c) (3, 1), (3,3), (4, 3), (4, 1).
d) (10, 1), (10, 3), (9, 3), (9, 1)
e) (12, 4), (13, 4), (13, 6), (12, 6)
(b) Join the points with coordinates:
(1, -5), (1, -1), (2, 0), (5, 0), (6, -1), (6, -5)
Sub-topic 4.2: Plotting Polygons (shapes)
Here we look at polygons plotted on coordinate axes, but first, recall
the names of polygons.
Names of polygons
Number of sides Name
3
4
5
6
7
8
9
10
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
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Note:
In a regular polygon:
(a) all the sides are the same.
(b) all the angles are of the same size.
Activity 4.2: The line AB is one side of a square
What are the possible coordinates of the corners of the square?
Exercise
1. In each case the coordinates of 3 corners of a square are given.
Find the coordinates of the other corner.
(a) (2, -2), (2, 3) and (-3, 3)
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(b) (2, 3), (3, 4) and (1, 4)
(c ) (2, 2), (4, 4) and (4, 0)
(d) (-6, 2), (-5, -5) and (1, 3)
(e) (-5, -2), (-2, -1), and (-1, -4)
2. The coordinates of 3 corners of a rectangle are given below. Find
the coordinates of the other corner of each rectangle.
(a) (-4, 2), (-4, 1) and (6, 1)
(b) (0, 2), (-2, 0) and (4, -6)
(c ) (-4, 5), (-2, -1) and (1, 0)
(d) (-5, 1), (-2, 5) and (6, -1)
3. (a) The coordinates of 2 corners of a square are (-4, 4) and (1, -1).
Explain why it is possible to draw three different squares using
these two points.
(b) Draw the three different squares.
(c ) If the coordinates of the corners had been (-5, 1) and (1, 3)
would it still be possible to draw 3 squares? Draw the possible
squares.
4. Half of a heptagon with one line of symmetry can be drawn by
joining the points with coordinates: (2, 4), (-2, 1), (-2, -1), (0, -3) and
(2, -3). Join the coordinates. You have drawn one half of the
heptagon. Complete the heptagon. Write down the coordinates.
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Sub-topic 4.3: Use of Appropriate Scale for Given
Data
Activity4.3: Plot the following points on the axes: (5, 50), (10,100),
(15,150), (20,200), (35, 350)
Do you realise that on the horizontal axis there are 5 units for each
space?
On the vertical axis there are 50 units for each space. So, what is the
scale for the axes?
Exercise
1. For each part, draw a pair of axes with suitable scales and plot the
points:
(a) (1, 15); (4, 35); (8, 45)
(b) (15, 100); (35, 500); (40, 700)
2. Plot the points (2, 60); (4, 50); (0, 70); (7, 60)
Situation of Integration
A Senior One learner has reported in her class and has settled at her
desk.
Support: The classroom is arranged in rows and columns. It is big a big
class with each learner having his/ her own desk.
Resources: Knowledge of horizontal and vertical lines i.e. rows and
columns, coordinates
Knowledge: counting numbers
Task: The mathematics teacher has asked her to explain how she can
access her seat, starting from the entrance of the class. Discuss
whether there are other ways of reaching her seat.
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Topic 5:
GEOMETRIC CONSTRUCTION SKILLS
Key Words: perpendicular lines, parallel lines, circumcircle, arcs
By the end of this topic, you should be able to:
i) draw perpendicular and parallel lines.
ii) construct perpendiculars, angle bisectors, mediators and
parallel lines.
iii) use compass and a ruler to construct special angles (600, 450).
iv) describe a locus.
v) relate parallel lines, perpendicular bisector, angle bisector,
straight line and a circle as loci.
vi) draw polygons.
vii) measure lines and angles.
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viii) construct geometrical figures such as triangle, square, rectangle,
rhombus, parallelogram.
Introduction
In this topic you will learn how to construct lines, angles and geometric
figures. Skills developed from this topic can be applied in day-to-day
life.
Sub-topic 5.1: Draw perpendicular and parallel
lines
Activity 5.1: Drawing perpendicular and parallel lines
(a) In your groups, list objects in real-life situations that can be used to
draw lines.
(b) Use the objects in (a) above to draw perpendicular lines, parallel
lines and intersecting lines.
Activity 5.2: Identifying lines
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In your groups, take a sheet of paper; divide it into half, then into half
again in the same way. Now fold your paper again. What kind of lines
do you see?
Next, fold the same paper into half in the opposite direction. Unfold
your paper now.
How is the new line you have created, related to the previous lines?
In real-life situations, where do we come across perpendicular lines
and parallel lines?
Which letters in the alphabet have the above lines?
In this sub-topic, you will have more hands-on work on perpendicular
and parallel lines
Sub-topic 5.2: Construction of Perpendicular
Lines
Activity 5.3: Construction of perpendicular line from an external
point to a given line
In your groups, work in pairs.
Given line segment AB and point C outside the line, construct a
perpendicular line from point C to line AB.
Taking the centre as C and any radius, draw two arcs on line AB at x
and y.
Now taking x as the centre and any radius, draw an arc below or above
the line opposite point C without changing the radius. Taking y as the
centre, draw an arc to intersect the previous arc. Join the intersection
of the arcs to point C .Compare your answers and make notes.
Activity 5.4: Construction of a Perpendicular line to a given point
on a given line segment
In your groups, work in pairs.
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Given line PQ and point Z on PQ. Taking Z as the centre and any
radius, draw two arcs on either side of Z name the arcs x and y . Now
taking x as the centre and any radius draw an arc either above or below
the line, without changing the radius now taking y as the centre draw
an arc to meet the previous arc join the intersection of the arcs to
point Z . Compare your answers with other group members.
Activity 5.5: Construction of a Perpendicular Bisector
In your groups work as an individual.
Given line segment AB. Taking A as centre and AB as the radius, draw
two arcs below and above the line, then now taking B as the centre and
without changing the radius, draw arcs to meet the previous arcs. Join
the intersection of the arc. What do you notice? Compare your work
with your group members.
Activity 5.6: Construction of parallel lines
In your groups, work in pairs.
Given line AB and point C outside the line. Take C as the centre, draw
an arc at point A taking AB as radius and A as the centre, draw an arc at
point B. Now take radius AC and taking B as the centre, draw an arc
above B, then taking radius AB and C as the centre, draw an arc to
meet the previous arc at D. Join the intersection of the arcs (D) to point
C. What do you notice. Name and describe shape ABCD. Compare your
answer with members of the your group.
Sub –topic 5.2: Using a Ruler, Pencil and Pair of
Compasses, Construct Special Angles
Activity5.7: Construction of special angles
In pairs, construct the following angles: 90o, 45o, 15o, 30o, 60o, 120o, 75o.
In your groups, compare your answers.
Using a protractor, measure your angles.
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Sub-topic 5.3: Describing Locus Question
What is the path traced out by the tip of the seconds-hand of a clock in
the course of each minute?
Activity 5.8: Discovering what Locus is
In your groups, discuss what happens if a goat is tied to a rope of
length 4 metres and around the place where the goat is, there are
gardens at a distance of 5 metres.
In your groups, draw sketches of the area where the goat can feed
from.
In real-life situations, where are such scenarios applied?
Activity5.8: Sketching and Describing Loci
In your groups, sketch and describe what happens about the following:
a) A mark on the floor as the door opens and closes.
b) The centre of a bicycle wheel as the bicycle travels along a
straight line.
c) A man is walking and keeping the same distance from two trees
P and Q.
d) A student is walking in the assembly hall keeping the same
distance from two opposite walls.
e) Compare your answers with other groups.
5.3.1: Relating Lines and Angles to Loci
According to the activities above, Locus is a trace of a point under
some conditions.
Activity5.9: Demonstration of some simple Loci
a) In your groups, demonstrate how one can walk the same distance
from a given point.
b) How one can walk the same distance from two fixed points.
c) How one can walk the same distance from a line.
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d) How one can walk the same distance from two intersecting lines.
In your different groups compare your answers.
Exercise
1. Construct the locus of a point equidistant from a fixed point.
2. Construct a locus of a point equidistant from a given line.
3. Construct the locus of a point equidistant from two intersecting
lines.
4. Construct a triangle ABC where AB = 12cm, AC=9cm and Angle BAC
= 60o. Find the point with the triangle where the distance from that
point to all the vertices of the triangle is equal taking that point as
the centre and the distance from the centre to the vertices as the
radius draw a circle. (vi, vii are implied.)
Sub-topic 5.4: Construction of Geometric Figure
Construction of geometric figures most of the time is application of
locus.
Activity5.10: Construction of geometrical figures
In pairs, construct a perpendicular bisector of any line segment.
Measure the distance from the perpendicular line to any of the points
on either side of the perpendicular bisector. What have you found out?
In your groups, construct an equilateral triangle with length 6cm.
Construct a circumcircle of the triangle. What type of locus is applied
here?
Exercise
1. Construct a triangle ABC in which AB = 8.5, BC = 6cm and angle B =
30o.
Construct a circle through the vertices of the triangle. Work out the
area of the circle.
2. Construct triangle PQR with PQ = QR= 7cm angle Q = 45o. Construct
a circumcircle of the triangle.
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3. Construct a parallelogram ABCD in which AB=5cm, BC=4cm and
angle B is 120o.
4. Construct an equilateral triangle ABC of sides 7cm.Bisect AB and BC
and let the bisectors intersect at X. With X as the centre and radius
XA, draw a circle.
Situation of Integration
In a village, there is an old man who wants to construct a rectangular
small house of wattle and mud.
Support: A string, sticks, panga, tape measure and human resource.
Resources: Knowledge of horizontal and vertical lines i.e. rows and
columns, knowledge of construction of geometric figures.
Task: The community asks you to accurately construct the foundation
plan for this old man’s house.
Explain to the class how you have accurately constructed the
foundation plan. Discuss whether there are other ways of constructing
an accurate foundation plan.
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Topic 6:
SEQUENCE AND PATTERNS
By the end of this topic, you should be able to:
i) draw and identify the patterns.
ii) describe a general rule of a given pattern.
iii) describe a sequence.
iv) determine a term in a sequence.
v) find the missing numbers in a given sequence.
Introduction
In this topic you will learn how to identify and describe general rules
for patterns. You will be able to determine a term in the sequence and
find the missing numbers in the sequence.
Sub-topic 6.1: Draw and Identify the Patterns
Activity 6.1: Identifying number pattens
In groups, work in pairs.
Look at the following sequences, how can you get the next number?
Compare your answers with other members.
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i) 3, 6, 9, 12, 15, …
ii) 2, 4, 6, 8, 10, 12, …
In (i), in order to get the next number, you add 2 to the previous
number. The numbers in this sequence are multiples of 3.
Sequence (ii), represents the multiples of 2.
Exercise
State the multiples of 3 found in this table:
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
This square shows multiples of a number. What is this number?
Write down the numbers that should go in each of these boxes. The
number square will help you with some of them.
a) The fifth multiple of … is …
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b) The …th multiple of … is 36
c) The 12th multiple of … is …
d) The 20th multiple of … is …
e) The …th multiple of … is 96.
f) The 100th multiple of … is …
Solution
a) the 5th multiple of 4 is 20
b) the 9th multiple of 4 is 36
c) the 12th multiple of 4 is 48
d) the 20th multiple of 4 is 80
e) the 24th multiple of 4 is 96
f) the 100th multiple of 4 is 400
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Exercise
1. On a number square like this one, shade all the multiples of 6. Then
answer the questions after the table.
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
a) What is the 4th multiple of 6?
b) What is the 10th multiple of 6?
c) What is the 12th multiple of 6?
d) What is the 100th multiple of 6?
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2. The multiples of a number have been shaded on this square. What
is the number?
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
Copy each statement about these multiples and write down the
numbers that go in the spaces.
a) The 3rd multiple of … is …
b) The 9th multiple of … is …
c) The 200th multiple of … is …
d) The …th multiple of … is 66
e) The … th multiple of … is 330.
3.
a) Write down the first 8 multiples of 8.
b) Write down the first 8 multiples of 6.
c) What is the smallest number that is a multiple of both 6 and 8?
d) What are the next two numbers that are multiples of both 6 and
8?
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4. a) Write down the first 6 multiples of 12.
b) What is the 10th multiple of 12?
c) What is the 100th multiple of 12?
d) What is the 500th multiple of 12?
e) If 48 is the nth multiple of 12, what is n?
f) If 96 is the nth multiple of 12, what is n ?
5. a) What multiples have been shaded in this number square?
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
b) What is the first multiple not shown in the number square?
6. a) Explain why 12 is a multiple of 6 and 4.
b) Is 12 a multiple of any other numbers?
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7. The number 24 is a multiple of 2 and a multiple of 3. What other
numbers is it a multiple of?
8. Two multiples of a number have been shaded on this number
square. What is the number?
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
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9. Two multiples of a number have been shaded on this number
square
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
a) What is the number?
b) What is the 19th multiple of this number?
10. Three multiples of a number are 34, 170 and 255. What is the
number?
11. Three multiples of a number are 38, 95 and 133. What is the
number?
12. Four multiples of a number are 49, 77, 133 and 203. What is the
number?
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Sub-topic 6.2: Describing the General Rule
Activity 6.2: Finding the Next Term
In your, groups work in pairs.
Can you use the given numbers of the sequence to deduce the pattern
and hence find the next term?
Example: What are the next 3 numbers in the sequence:
a) 12, 17, 22 …?
b) 50, 47, 44, 41, 38, …?
Compare your answers with other group members
Solution
a) To find the pattern, it is usually helpful to first find the differences
between each term i.e. the difference between 12 and 17 is 5; the
difference between 17 and 22 is 5.
So the next term is found by adding 5 to the previous term. This gives
you 27, 32, 37.
b) Again you find the difference between:
i) 50 and 47 is -3.
ii) 47 and 44 is -3.
iii) 44 and 41 is -3.
iv) 41 and 38 is -3.
So, the next term is found by taking away 3 from the previous term,
giving you 35, 32, 29.
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Exercise
1. Copy the following exercise and find the sequence in each case,
giving the next three numbers.
a) 18, 30, 42, 54, 66, …
b) 4.1, 4.7, 5.3, 5.9, 6.5, …
c) 8, 14, 20,…, 32, …
d) 3, 11, …, 27, 35, …
e) 3.42, 3.56, 3.70, 3.84, 3.98, …
f) 10, 9.5, 9, 8.5, 8, 7.5, …
2. Copy each sequence and fill in the missing numbers.
a) 2, 4, …, 16, 32, …
b) 100, 81, 64, …, 36, …
c) 6, 9, …, 21, 30, 30, …
d) 0, 1.5, 4, …, 12, …
e) 1, 7, 17, …, 49, …
Sub-topic 6.3: Generating Number Sequence
Activity 6.3: Generating a sequence
In your groups work in pairs.
You can use formulae to generate sequences. For example, the formula
5n, with n = 1, 2, 3, 4, … generates the sequence 5x1, 5x2, 5x3, 5x4, …
The sequence generated is 5, 10, 15, 20, …
Example: What sequence do you generate by using the following
formula?
a) 5n – 1
b) 6n + 2
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Solution
a) putting n = 1, 2, 3, 4, … gives 4, 9, 14, 19, …
b) putting n = 1, 2, 3, 4, … gives 8, 14, 20, 26, …
You can find the formula for this sequence, 11, 21, 31, 41, 51, 61, …
How you can find the sequence. The sequence begins with 11, and 11 =
10 + 1. Continue to add 10 each time the formula is 10n + 1.
Compare your answers with other members in the group.
Exercise
1. What number comes out of each of these number machines?
2. The sequence 1, 2, 3, 4, 5, … is put into each number machine. What
does each machine do?
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3. Write down the first 5 terms of the sequence given by each of these
formulae:
a) 9n b) 12n c) 2n + 4 d) 3n – 1 e) 3n - 2
4. a) What is the 10th term of the sequence 2n + 1?
b) What is the 8th term of the sequence 3n + 6?
c) What is the 5th term of the sequence 4n + 1?
d) What is the 7th term of the sequence 5n – 1 ?
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5. Draw double machines that could be used to get each of these
sequences from 1, 2, 3, 4, 5 …
Also write down the formula for each sequence of the following:
a) 5, 9, 13, 17, 21, …
b) 2, 5, 8, 11, 14, …
c) 6, 11, 16, 21, 26, …
d) 4, 9, 14, 19, 24, …
e) 102, 202, 302, 402, 502, …
Sub-topic 6.4: Formulae for General Terms
Activity 6.4 : Identifying the nth term
In your groups work in pairs.
Note: It is very helpful not only to be able to write down the next few
terms in a sequence, but also to be able to write down, for example,
the 100th or even the 1000th term.
Example: For the sequence 3, 7, 11, 15, …, …
Find:
a) the next three terms.
b) the 100th term.
c) the 1000th term.
Answer
a) You can see that 4 is added each time to get the next
term.
So you obtain 19, 23, 27.
b) To find the 100th term, starting at 3, you add 3 to 4 times ninety
nine times giving
3 + 4 x 99 = 3 + 396 = 399
c) Similarly, the 1000th term is
3 + 4 x 999 = 3 + 3996 = 3999
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I can go one step further and write down the formula for a general
term, i.e. the nth term.
This is 3 + 4 x (n – 1) = 3 + 4n - 4
= 4n – 1.
Compare your answers with other members of the group and the
examples given.
Exercise
1. For each sequence, write down the difference between each term
and formula for the nth term.
a) 3, 5, 7, 9, 11, …
b) 5, 11, 17, 23, 29, …
c) 4, 7, 10, 13, 16, …
d) 2, 5, 8, 11, 14, …
e) 6, 10, 14, 18, 22, …
2. a) Write down the first 6 multiples of 11.
b) What is the formula for the nth term of the sequence of the
multiples of 11?
c) What is the formula for the nth term of this sequence?
3. The formula for the nth term of this sequence is n2.
1, 4, 9, 16, 25, …
What is the formula for the nth term of the following sequences?
a) 0, 3, 8, 15, 24, …
b) 10, 13, 18, 25, 34,
c) 2, 8, 18, 32, 50, …
d) 1, 8, 27, 64, 125, …
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Situation of Integration
There is a family in the neighbourhood of your school. The family has a
rectangular compound on which they want to put up a hedge around.
The hedge shall be made up of plants of different colours.
Support: Physical instruments like hoes, machetes, tape measure
Resources: Knowledge of construction of figures like rectangles,
patterns, sequences
Task: The family requests you to plant the hedge around their
rectangular compound so that it looks beautiful.
Explain how you will plant the hedge, making sure that the plants at
the corners of the compound are the same in terms of colour.
Discuss whether there are other ways of planting the hedge.
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Topic 7:
BEARINGS
The diagram below shows the bearing of Kabale from where the lady is
standing.
Key words: angle, direction, bearing, scale, line, turn
By the end of this topic, you should be able to:
i) review the compass.
ii) describe the direction of a place from a given point using
cardinal points.
iii) describe the bearing of a place from a given point.
iv) draw suitable sketches from the given information.
v) choose and use appropriate scale to draw an accurate drawing.
v i) differentiate between a sketch and a scale drawing.
vii) apply bearings in real life situations.
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Introduction
In this topic you will learn how to tell the bearing of a point from a
given point. You will determine accurately the distance between two
points.
Sub-topic 7.1: Angles and Turns
You will need to understand clearly, what the terms such as turn, halfturn,
etc. mean in terms of angles. There are 360o in one complete turn,
so the following are true.
You also need to refer to compass points: north (N), south(S), east(E),
west(W), northeast (NE), southeast (SE), southwest (SW) and northwest
(NW)
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Activity 7.1: Identifying the angles in relation to the compass
direction
Work in pairs
Do the following turns and in each case state the size of the angle you
have turned through.
i) Turn from N to S clockwise or anticlockwise
ii) Turn from NE to SE clockwise
iii) Turning clockwise from NE to E
Example
What angle do you turn through if you turn:
(a) from NE to NW anticlockwise?
(b) from E to N clockwise?
Compare your answers with the rest of the members in class.
Solution
(a) 90o (or ¼ turn)
(b) 270o (¾ turn)
Exercise
1. What angle do you turn through if you turn clockwise from:
(a) N to E? (b) W to NW? (c) SE to NW? (d) NE to N? (e) W to NE?
(f) S to SW? (g) S to SE? (h) SE to SW? (i) E to SW?
2. In what direction will you be facing if you turn:
(a) 180o clockwise from NE?
(b) 180o anticlockwise from SE?
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(c ) 90o clockwise from SW?
(d) 45o clockwise from N?
(e) 225o clockwise from SW?
(f) 135o anticlockwise from N?
(g) 315o clockwise from SW?
3. The sails of a windmill complete one full turn every 40 seconds.
(a) How long does it take the sails to turn through:
(i) 180o (ii) 90o (iii) 45o?
(b) What angle do the sails turn through in:
(i) 30 seconds? (ii) 15seconds? (iii) 25 seconds?
Sub-topic 7.2: Bearings
The bearing of a point is the number of degrees in the angle measured
in a clockwise direction from North line to the line joining the centre of
the compass with the point. A bearing is used to present the direction
of one-point relative to another point.
Activity7.2: Estimating bearings of some places within the school
compound
In groups, work in pairs and outside the classroom.
From your school flag post, estimate the bearings of each building
found in the School.
Note: Three figures are used to give bearings.
All bearings are measured in a horizontal plane.
Compare your answers with the other members of the group.
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Exercise
1. Find the bearing of each of the following directions:
(a) S (b) NE (c) N (d) NW
2. Find the bearing of each of the following directions:
(a) N600E (b) N350E (c) N900W (d) S400E
3. Draw a scale diagram to show the position of a ship which is 270 km
away from a port on a bearing of 110o.
Situation of Integration
Ajok is in Kampala City and has been told to use a car to move to Lira
town. She has never gone to Lira. She has been given the map of
Uganda showing routes through which she can access Lira town.
Support: Mathematical instruments, pencil, paper, pens, tracing paper
and map of Uganda
Resources: Knowledge of construction of figures like triangles, lengths
of sides of triangles, operations on numbers.
Task: Ajok wants to use the short distance from Kampala to Lira.
Explain how Ajok can determine the shortest distance. Using the map
given to her is it possible for Ajok to use the shortest distance she has
determined. Explain your answer.
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Topic 8:
GENERAL AND ANGLE PROPERTIES OF
GEOMETRIC FIGURES
Key words: line segment, transversal, parallel
By the end of this topic, you should be able to:
a. identify different angles.
b. solve problems involving angles on a straight line, angles on
transversal and parallel lines.
c. state and use angle properties of polygons in solving problems.
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Introduction
In bearings you studied angle turns, and in this topic you will study
angles on the straight line, parallel lines and angle properties of
polygons. Equipped with the knowledge from this topic, you will be
able to solve problems related with angle properties.
You will need to understand clearly what the terms such as turn, halfturn,
etc. mean in terms of angles. There are 360o in one complete turn,
so the following are true.
i) Turning from N to S is 180o clockwise or anticlockwise.
ii) Turning from NE to SE is 90o clockwise (or 270o anticlockwise).
iii) Turning clockwise from NE to E is 45o (or 315o anticlockwise).
Example
What angle do you turn through if you turn:
a) from NE to NW anticlockwise?
b) from E to N clockwise?
Solution
c) 90o (or ¼ turn)
d) 270o (¾ turn)
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Sub-topic 8.1: Identify Different Angles
Activity8.1: Identifying objects that form angles
In your groups, work in pairs.
Identify objects in you class, which make 900, 1800, 3600
A protractor can be used to measure angles.
Note:
The angle around the circle is 360o.
The angle around a point on a line is 180o.
A right angle is 90o
Compare your answers with other members of the group and
classify them
Exercise
1. For each of the following angles, first estimate the angles and then
measure the angle to see how good your estimate was.
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2. Draw the following angles
(a) 20o (b) 42o (c ) 80o (d) 105o (e) 170o (f) 200o (g) 275o (h) 305o
3. Immaculate finds out the favourite sports for members of her class.
She works out the angles in the list shown below for a pie chart. Draw
the pie chart.
Sport Angle
Football
Swimming
Tennis
Rugby
Hockey
Badminton
Other
1100
70o
80o
40o
30o
10o
20o
Exercise
1. (a) Draw a triangle with one obtuse angle.
(b) Draw a triangle with no obtuse angles.
2. Draw a four-sided shape with:
a) one reflex angle.
b) two obtuse angles.
Sub- topic 8.2: Angles on a Line and Angles at a
Point
Remember that:
a) angles on a line add up to 180o
And:
b) angles at a point add up to 360o
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These are two important results, which help when finding the size of
unknown angles.
Activity 8.2: Identifying angles
Work as individuals
Draw two intersecting lines. Use your mathematical instruments to
measure the angles formed at the intersecting point.
i) How many angles have been formed at the point of
intersection?
ii) What is the size of each angle formed?
Compare your work with your friends and note your findings.
A polygon is a closed plane figure with straight sides.
Activity 8.3: Identifying the polygons
In pairs:
Find the number of sides of different polygons and their corresponding
names. Also determine the number and size of interior and exterior
angles of the regular polygons.
Compare your answers with other members’.
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Exercise
1. If the vertices of a regular hexagon are joined to the centre of
the hexagon, what is the size of each of the six angles at the
centre? Use your answer to construct a regular hexagon
ABCDEF of side 3cm. Start with a circle of radius 3cm. Measure
the length of the diagonal AC.
2. Find the sum of the interior angles of a polygon with 22 sides.
3. The interior angle of a regular polygon is 1620. How many sides
has the polygon?
Activity of Integration
A diagram of a table showing coffee production in Uganda from year
2015 to year 2019
Year 2015 2016 2017 2018 2019
Production
(tonnes)
20 23 18 30 49
Task: The chairperson of Karo Farmers Association was asked to
represent the information above on pie chart. As a senior one
learner help him solve the challenge.
Support: Mathematical set
Resource: Knowledge of angles
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Topic 9:
DATA COLLECTION AND
PRESENTATION
Key words: data, chart, pie, quantitative, qualitative, discrete,
continuous, hypothesis
By the end of this topic, you should be able to:
a) understand the differences between types of data.
b) collect and represent simple data from local environment using
bar chart, pie chart and line graph.
Introduction
In this topic, you will learn different types of data, data collection,
presentation and analysis.
Sub-topic 9.1: Types of Data
Qualitative data is data that is not given numerically; e.g. favourite
colour, place of birth, favourite food, and type of car.
Quantitative data is numerical. There are two types of quantitative
data: discrete and continuous data. Discrete data can only take
specific numeric values e. g. shoe size, number of brothers, number of
cars in a car park. Continuous data can take any numerical value e.g.
height, mass, length.
Activity 9.1: Identifying types of data
In your groups identify which of the following terms best describes
each of the information listed (i) to (vii)?
Give reasons for your response.
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• Qualitative data
• Continuous Quantitative Data
• Discrete Quantitative Data
i) Age
ii) Birth place
iii) Height
iv) World Ranking
v) Aces
vi) First serve School
vii) School life
In your groups identify more examples.
Exercise
1. Mr Okot starts to make a database for his lesson.
Name Age Primary school Transport
to School
Height Reading Glasses
Alice 11 St. Johns Bus 145cm yes
Ben 12 St. Andrews Walk 160 cm no
Carol 12 Hilltop Car 161 cm no
David 12 Hilltop 152 cm no
Eddie 11 St. Andrews Walk 158 cm yes
Fredrick St. Andrews Bike 164 cm no
Graham 12 St. Johns Bus 166 cm yes
a) What is missing from Mr Okot’s database?
b) Which columns in the database contain quantitative data?
c) Which columns in the database contain qualitative data?
d) Write down what Mr Okot would put in his database if you joined
his class.
2. Which of the following would give:
(a) qualitative data
(b) discrète quantitative data
(c ) continuous quantitative data
(i) Mass (ii) Number of cars
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(iii) Favourite football team (iv) Colour of car
(v) Price of chocolate bars (vi) Amount of pocket money
(vii) Distance from home to school (viii) Number of pets
(ix) Number of sweets in a jar (x) Mass of crisps in a packet.
3. The table below shows a database that has no entries.
Name Age Favourite
food
Favourite
TV show
Favourite
pop group
Time spent
watching TV
yesterday
a) Collect data from 10 people to complete the data base.
b) State whether each column contains:
i) qualitative data.
ii) continuous quantitative data.
iii) or discrete quantitative data.
c) Answer the following questions:
i) What is the most popular TV show?
ii) Who is the oldest?
iii) What is the favourite pop group for the youngest person?
d) Write 3 more questions you could answer using your database and
write the answers to them.
Sub-topic 9.2: Collecting Data
In this section, you will see how data is collected, organized and
interpreted, using a tally chart and then displayed using:
i) Pictograms
ii) Bar charts
iii) Pie charts
Note:
A hypothesis is an idea that you want to investigate to see if it is true or
false. For example, you might think that most people in your school get
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there by bus. You could investigate this using a survey. A tally chart can
be used to record your data.
Activity 9.2: Collecting data
In groups identify the means of transport each learner use to come to
school. As a class identify how many of you use the same means of
transport.
i) Which means of transport is used by the majority?
ii) Which one is the least used means of transport?
Example
The learners in a class were asked how they got to school.
Method of Travel Tally Frequency
Walk ///// /// 9
Bike /// 3
Car ///// / 6
Bus ///// ///// // 12
TOTAL 30
Illustrate this data using:
a) a pictogram
b) a bar chart
c) a pie chart
What are the main conclusions that can be deduced from the data?
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Solution
(a) If (stick man) is taken to represent 2 people, then the pictogram
looks like:
i) Walk (4 and a half stick men)
ii) Bike (1 and a half stick men)
iii) Car (3 stick men)
iv) Bus (6 stick men)
(b) A bar chart for the data is illustrated below:
(c) To illustrate the data with a pie chart, you need to find out what
angle is equivalent to one pupil. Since there are 360o in a circle and
30 pupils, then angle per pupil is 360 ÷ 30 = 12o.
To find the angle for walk, when there are 9 pupils, it is simply:
9 x 12 = 1080
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The complete calculations are shown below:
Method of
travel
Frequency Calculation Angle
Walk 9 9 x (360 ÷ 30) 108o
Bike 3 3 x(360 ÷ 30) 36o
Car 6 6 x (360 ÷ 30) 72o
Bus 12 12 x (360 ÷ 30) 144o
TOTAL 360o
The corresponding pie chart is shown below:
From the data we can see that:
• the most common way of getting to school is by bus. (This is
called the modal class or the mode.)
• the least popular way of getting to school is by bike.
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Exercise
1. The children in a class were asked to state their favourite crisps.
The results are given in the tally chart below:
Flavour Tally Frequency
Ready Salted /////
Salt and Vinegar ///// ////
Cheese and onion ///// //
Prawn Cocktail ///
Smokey Bacon ///// /
TOTAL
(a) Copy and complete the table by filling the frequencies.
(b) Represent the data on a bar chart.
(c) Draw a pictogram for this data.
(d) Copy and complete the following table and draw a pie chart.
Flavour Frequency Calculation Angle
Ready Salted 5 5 x ( 360o ÷ 30) 60o
TOTAL
(e) What flavour is the mode?
2. (a) Do you think salt and vinegar crisps will be most popular crisps in
your class?
(b) Carry out a favourite crisps survey for your class. Present the
results in a bar chart and state which flavour is the mode.
(c) Was your hypothesis in (a) correct?
3. “Most children in my class are 1.3m tall.”
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(a) Collect data to test this hypothesis.
(b) Present your data in a suitable diagram.
(c) Was the original hypothesis correct?
4. Is the music group that is most popular with the boys in your class
the same as the music group that is most popular with girls?
(a) Write down a hypothesis that will enable you to answer this
question.
(b) Collect suitable data from your class.
(c) Present your data using a suitable diagram.
(d) Was the hypothesis correct?
Situation of Integration
The Games Master at your school wants to buy football boots for the
three teams in the school. The three teams are the under 18 years,
under 16 years and the under 14 years. The Games Master does not
know the foot size for each of the players.
Support: pens, paper, tape measure, team members
Resources: Knowledge of tabulation, of tallying, of approximation, of
central measures and of collection of suitable data.
Task: The total number of players for the three teams is 54. The Games
Master wants to know the size of the boots for each player and
the number of pairs for each size.
Explain how the Games Master will get the required data and how to
determine the total cost for buying the football boots for the 54
players.
Is there another way of getting the required data other than what you
have explained above?
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Topic 10:
REFLECTION
By the end of this topic, you should be able to:
i) identify lines and planes of symmetry for different figures.
ii) state and use properties of reflection as a transformation.
iii) make geometrical deductions using reflection (distinguish
between direct and opposite congruence).
iv) apply reflection in the Cartesian plane.
Introduction
In this topic, you will learn how to identify the lines of symmetry, state
the properties of reflection as a transformation, make geometrical
deductions and apply reflection in Cartesian plane.
The image of a figure by reflection is its mirror image in the axis or
plane of reflection. For example the mirror image of the letter p for
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reflection with respect to a vertical axis would look like q. Its image by
reflection in a horizontal axis would look like b.
Sub-topic 10.1: Identify Lines of Symmetry for
Different Figures
Activity 10.1: Identifying lines of symmetry
In pairs:
1. Fold a piece of paper in half
2. Open the paper and put in one drop of ink on the fold
3. Close the paper over the ink and press down hard on the paper.
4. When the ink has dried, open up your paper.
(a) Look at both sides of the fold line. Are they the same size and
shape?
(b) Look at any two corresponding points on the ink blot, one on
either sides of the fold.
(i) What can you say about the distance from one point to
the fold line and the distance from the corresponding
point to the fold line?
(ii) If a line joins two corresponding points, what is the angle
between the line and the fold?
Exercise 1
1. Draw a rectangle on a tracing paper. Fold it to find the lines of
symmetry. How many lines of symmetry does a rectangle have?
2. Find the number of lines of symmetry of (a) a square (b) an
equilateral triangle (c) a rhombus
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Sub-topic 10.2: Reflection in the Cartesian Plane
Activity 10.2: Reflecting in a Cartesian plane
In your groups, work as pairs.
Plot the points P (5, 4), Q (-1, 3) and R (0, -2) on squared paper.
a) A mirror is placed on the x axis. Where would the images of the
tree points be?
What are the coordinates of the image points P’, Q’ and R’?
b) Draw another pair of axes. Plot the same points again. Take the
line y = 2 as the mirror line. Where would the images of the
three points be? What are the coordinates of the new image
points P’, Q’ and R’?
c) Draw another pair of axes. Draw the line x = 4. Plot the points (1,
-3). Using the line x = 4 as the mirror line, find the image of the
point (1, -3).
Compare your answers with other members in your group.
Exercise 2
1. Find the image of the point (2, 5) under reflection in the y axis.
2. After a point has been reflected in the x axis, its image is at (3,
2). Find the coordinates of the object point.
3. The points A(4, 2) , B(1, 3) and C(1,-2)are reflected in the line y =
x. Find the coordinates of A’ , B’ and C’, the images of A and B.
Situation of Integration
One of your relatives wants to make a barbershop /hairdresser. He
approaches you for help.
As a senior one graduate draw a plan of how you can help your relative
make his /her barber shop be up to date.
Support: Interior plan of the shop
Task: Advice the barber to make sure the customers can view
themselves with their images not distorted.
Resource: knowledge of reflection
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Topic 11: Equation of Lines and
Curves
Key words: variable, curve, substitution
By the end of this topic, you should be able to:
i) form linear equations with given points.
ii) draw the graph of a line given its equation.
iii) differentiate between a line and a curve.
Introduction
In this topic you will tell the difference between a line and a curve, how
to form linear equations and draw graphs for the given linear
equations.
Sub-topic 11.1: Fundamental Algebraic Skills
In this section, you will look at some fundamental algebraic skills by
examining codes and how to use formulae.
Example
If a = 4, b = 7 and c = 3, calculate:
(a) 6 + b (b) 2a + b (c) ab (d) a (b – c) (d) a (b – c)
Solution
(a) 6 + b = 6 + 7 = 13
(b) 2a + b = 2x4 + 7 = 8 + 7 = 15
(c) ab = 4x7 = 28
(d) a (b – c) = 4 x (7 – 3) = 4 x 4 = 16
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Example
Simplify where possible:
(a) 2x + 4x (b) 5p + 7q – 3p + 2q
(c) y + 8y – 5y (d) 3t + 4s
Solution
(a) 2x + 4x = 6x
(b) 5p + 7q – 3p + 2q = 5p -3p + 7q+2q = 2p – 9q
(c) y + 8y – 5y = 9y – 5y = 4y
(d) 3t + 4s = 3t + 4s
Exercise
1. If a = 2; b = 6; c = 10 and d = 3, calculate:
(a) a + b (b) c – b (c) d + 7
(d) 3a + d (e) 4a (f) ad
(g) 3b (h) 2c (i) 3c - b
(j) 6a + b (k) 3a + 2b (l) 4a – d
2. If a = 3; b = -1; c = 2 and d = -4, calculate:
(a) a – b (b) a + d (c) b + d
(d) b – d (e) 3d (f) 5(d – c)
(g) a (b + c) (h) d(b + a) (i) c(b – a)
(j) a (2b – c) (k) d(2a – 3b) (l) c(d – 2)
3. Simplify, where possible:
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(a) 2a + 3a (b) 5b + 8b
(c) 6c – 4c (d) 5d + 4d + 7d
(e) 6e + 9e – 5e (f) 8f + 6f – 13f
(g) 9g + 7g – 8g - 2g -6g (h) 5p + 2h
(i) 3a + 4b – 2a (j) 6x + 3y – 2x –y
(k) 8t – 6t + 7s – 2s
(l) 11m +3n – 5p + 2q -2n +9q -8m + 14p
4. Sam asks his friend to think of a number, multiply it with 2 and then
add 5. If the number his friend starts with is x, write down a formula for
the number her friend gets.
Subtopic 11.2: Function Machines
In this section you will look at how to find the input and output of
function machines.
INPUT → FUNCTION MACHINE→ OUTPUT
Activity 11.1: Function machine activity
In pairs try out the numbers the first one is done for you.
Calculate the output of each of these function machines:
(a) 4 →x5 →?
(b) 5 →x2 →-1 →?
(c) -3 →+8→x7→?
(d) 8 →+6→x9→?
(e) -5→+3→x7→?
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Compare your answers with members of the group.
Solution
(a) The input is simply multiplied by 5 to give 20:
4 → x5 → 20
Exercise
1. What is the output of each of these function machines:
(a) 4→+6 →?
(b) 3 →x10→?
(c) 10→-7→?
(d) 14→÷2→?
(e) 21→÷3→?
(f) 100→×5→?
2. What is the output of each of these function machines:
(a) 3→×4→-7→?
(b) 10→-8→×7→?
(c) 8→-5→×5→?
(d) -2→×6→+20→?
(e) 7→+2→÷3→?
(f) -5→+8→×9→?
3. What is the input of each of these function machines:
(a)? →×5→30 (b)? → +8→ 12
(c)? → -9→ 11 (d)? → +4→ 5
(e)? → +12→ 21 (f)? → ×7→ 42
4. A number is multiplied by 10, and then 6 is added to get 36. What is
the number?
5. Karen asked her teacher, Maria, how old she was. The teacher
replied that if she double her age, added 7 and then divided by 3, she
would get 21. How old is Karen’s teacher?
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6. A bus has a maximum number of passengers when it leaves the bus
station. At first stop, half of passengers alighted. At the next stop 7
people alighted and at the next stop 16 people alighted. There are
now 17 people on the bus. How many passengers were on the bus
when it left the bus station?
Sub-topic 11.3: Linear Equations
An equation is a statement, such as 3x + 2 = 17, which contains an
unknown number. In this case, it is x. The aim of this section is to show
how to find the unknown number, x.
All equations contain an ‘‘equals” sign.
To solve the equation, you need to reorganize it so that the unknown
value is by itself on one side of the equation. This is done by
performing operations on the equation. When you do this, in order to
keep the equality of the sides, you must remember that “Whatever
you do to one side of an equation, you must also do the same to the
other side”.
Example
Solve these equations:
(a) x + 2 = 8 (b) x- 4 = 3 (c ) 3x = 12
(d) 2x + 5 = 11 (e) 3 – 2x = 7
Solution
(a) To solve this equation, subtract 2 from each side of the equation:
X + 2 = 8
X + 2 -2 = 8 – 2
X = 6
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(b) To solve this equation, add 4 to both sides of the equation:
X – 4 = 3
X – 4 + 4 = 3 + 4
X = 7
(c) To solve this equation, divide both sides of the equation by3:
3x = 12
3x ÷ 3 = 12 ÷ 3
X = 4
(d) This equation must be solved in 2 stages.
First, subtract 5 from both sides:
2x + 5 = 11
2x + 5 -5 = 11 – 5
2x = 6
Then, divide both sides of the equation by 2:
2x ÷ 2 = 6 ÷ 2
X = 3.
(e) First, subtract 3 from both sides:
3 – 2x = 7
3 – 3 – 2x = 7 – 3
-2x = 4
Then divide both sides by (-2);
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-2x ÷ -2 = 4 ÷ -2
X = -2.
Example 3
You ask a friend to think of a number. He then multiplies it by 5 and
subtracts 7. He gets the answer 43
a) Use this information to write down an equation for x, the unknown
number.
b) Solve your equation for x
Solution
a) As x = number your friend thought of, then
5x
X 5x So 5x -7 = 43
b) First, add 7 to both sides of the equation to give
5x = 50
Then divide both sides by 5 to give
X = 10
And this is the number that your friend thought of.
Exercises
1. Solve these equations:
a) x +2 = 8 b) x +5 = 11 c) x – 6 = 2
d) x – 4 = 3 e) 2x = 18 f) 3x = 24
x 5 -7
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g) 4
6
x = h) 9
5
x = i) 6x = 54
j) x + 12 = 10 k) x + 5 =3 l) x -22 = -4
m) -2
7
x =
n) 10x = 0
o) 5
2
x =
2. Solve these equations
a) 2x + 4 =14 b) 3x + 7 =25 c) 4x +2 = 22
d) 6x – 4 = 26 e) 5x – 3 = 32 f) 11x – 4 =29
g) 3x + = 4 = 25 h) 5x – 8 = 37 i) 6x + 7 = 31
j) 3x + 11 = 5 k) 6x + 2 = -10 l) 7x + 44 +2
3. Solve these equations, giving your answers as fractions or
mixed numbers
a) 3x = 4 b) 5x = 7 c) 2x + 8 = 13
d) 8x + 2 = 5 e) 2x +6 =9 f) 4x = 7 = 10
4. Solve these equations:
a) x + 2 =2x -1 b) 8x – 1 = 4x + 11 c) 5x + 2 = 6x - 4
d) 11x – 4 = 2x = 23 e) 5x +1 = 6x -8 f) 3x + 2 +5x + x =44
g) 6x + 2 – 2x = x + 23 h) 2X – 3 = 6x + x -58 i) 3x + 2 = x -8
j) 4x – 2= 2x - 8 k) 3x + 82 = 10 x + 12 l) 6x – 10 = 2x - 14
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Topic 14: Time and Time Tables
Key words: quarter, timetable, half, departure, arrival
By the end of this topic, you should be able to:
i) identify and use units of time.
ii) use and interpret different representations of time.
iii) apply the understanding of time in a range of relevant real life
contexts.
Introduction
In this topic, you will learn various units of time, such as minutes,
seconds, hours, day, week, month, year. You will be able to understand
and apply time in a range of relevant real-life contexts.
Sub-topic 14.1: Telling the Time
In this section we look at different ways of writing times; for example,
7:45 is the same time as quarter to eight.
On a clock face, this can be represented as shown below.
Also remember that
One hour = 60 minutes
So that
Half an hour = 30 minutes
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Quarter of an hour = 15 minutes
Three quarters of an hour = 45 minutes
Example
Write each time using digits and show the position of the hands on a
clock face:
(a) twenty-five past eight.
(b) quarter to ten.
Solution
(a) Twenty-five past eight using digits is 8:25
(b) Quarter to ten can be thought of as:
15 minutes to 10 o’clock
Or
45 minutes past 9 o’clock
So, using digits, quarter to ten is 9:45
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Exercise
1. Draw the time given below on clock faces:
(a) ten past five (b) ten minutes to nine (c ) quarter to seven
(d) quarter past twelve (e) half past ten (f) twenty nine minutes to five
(g) ten minutes to two (h) twenty five minutes to six (i) twenty past
four
2. Draw the following time on clock faces:
(a) 4:00 (b) 5:30 (c ) 7:15 (d) 8:20 (e) 2:45 (f) 3:50
(g) 1:55 (h) 6:05 (i) 11:35
3. Write the following time in words:
(a) 9:30 (b) 4:00 (c ) 4:25 (d) 8:45 (e) 7:35 (f) 9:05
4. Write these times using digits:
(a) eight o’clock (b) quarter to seven (c ) ten past five
(d) half past six (e) ten to three (f) five to four
(g) twenty five to nine (h) twenty to three
Sub-topic 14.2: 12-hour and 24-hour Clocks
The 24-hour clock system can be used to tell if time is in the morning or
the afternoon. Alternatively, time can be given as am or pm.
Activity 14.1: Converting from 12 hour to 24 hour and vice versa
In pairs:
i) Write these times in 24-hour clock time:
(a) 3:06 am (b) 8:14 pm (c) 9:45am (d) 3:06pm
ii) Write these times in 12-hour clock time:
(a) 03:00 (b) 09:45 (c) 13:07 (d) 22:15
Solution
(a) As this is a.m. the time remains the same except you add a zero in
front of 3, so the time becomes 0306 in a 24-hour clock.
(b) As this is pm, you add 12 to the hours to give you 2014 in a 24-hour
clock.
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Example
Write these times using am or pm in a 12-hour clock.
(a) 14:28 (b) 07:42
Solution
(a) As the hours figure, 14, is greater than 12, subtract 12 and write as a
pm time. The answer is2:28pm.
(b) As the hours figure, 07, is less than 12, simply remove the zero and
then write the time as am. The answer is 7:42 am.
Exercise
1. Convert the following time to the 24-hour clock:
(a) 9:24am (b) 11:28pm (c ) 11:14a.m (d) 7:13pm
2. Write the following time in the 24-hour clock:
(a) quarter to eight o’clock in the morning
(b) ten minutes to midnight
(c) ten past nine o’clock in the morning
(d) half past two o’clock in the afternoon
3. Write the following24-hour clock in words
(a) 14 :30 (b) 15:55 (c) 07:45
4. Sarah leaves home at 09:00 and returns 7 hours later. Write the time
that Sarah gets home in the 24-hour clock and in the twelve-hour
clock using am or pm.
Sub-topic 14.3: Units of Time
In this section we explore the different units of time.
1 minute = 60 seconds
1 hour = 60 minutes
1 day = 24 hours
1 week = 7 days
1 year = 365 0r 366 days
Example
1. How many hours are there in May?
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Solution
Number of hours in May = 31 x 24 = 744 hours
Activity 14.2
In pairs find out if 25 February is a Friday. What will be the date on the
next Friday?
(a) If it is not a leap year.
(b) If it is a leap year?
Compare your answers with members of the group before you check
the solution.
Solution
(a) You could write out the 7 days like this:
Friday 25
Saturday 26
Sunday 27
Monday 28
Tuesday 1
Wednesday 2
Thursday 3
Friday 4
Or
25 + 7 = 32
32 – 28 = 4
So the next Friday will be 4th March.
(b) Using the addition method:
25 + 7 = 32
32 – 29 = 3
So, in a leap year, the next Friday will be 3rd March.
Exercise
1. How many hours are there in a week?
2. How many hours are there in:
(a) September?
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MATHEMATICS
PROTOTYPE
121
(c) February?
(d) one year?
3. 3. Rupert goes on holiday on Monday 20th June. He returns 14
days later. On what date does he return from his holiday?
4. 4. If 3rd October is a Monday:
(a) What day of the week will 1st November be?
(b) What will be the date of the first Monday in November?
5. Immaculate goes to the bank every Tuesday. The last time she
went was on Tuesday 20th October.
(a) What will be the dates of her next 2 visits to the bank?
(b) On the second Tuesday in November she is ill and goes to the
bank on Wednesday instead. What is the date of that
Wednesday?
Sub-topic 14.4: Timetables
In this section we consider how to extract information from timetables.
Exercise
1. The table below gives the timetable for a Bus that runs from Mbale
to Kampala.
Mbale depart 08:57
Iganga depart 10:06
Jinja arrive 16:57
Mukono arrive 17:23
Kampala arrive 17:42
(a) At what time does the bus leave Mbale?
(b) At what time does the bus arrive at Kampala?
(c) Where does the bus arrive at 16:57?
(d) Mr Okot arrives in Mbale at five past nine. Can he catch the bus?
2. Mike is in Brussels and wants to return to Ashford. He looks at this
train timetable:
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SENIOR ONE
122
Brussels to Waterloo
Brussels Midi
Lille Europe
Ashford
International
Waterloo
International
0856
0937
0938
1047
1102
1142
1141
1247
1302
1342
1341
1447
1456
1536
1536
1639
1702
1742
1741
1843
1756
1836
1837
1939
1856
1936
1938
203
2102
2142
….
2239
a) At what time should he catch a train if he wants to arrive in
Ashford at 17:41?
b) Which train should he avoid if he wants to go to Ashford?
c) If he catches the 14:56 train, at what time does he arrive in
Ashford?
d) He catches the 14:56, but falls asleep and does not get off at
Ashford. At what time does he get to Waterloo?
3. The Journey from Kabale (Uganda) to Kigali (Rwanda) takes 2 ½
hours. The time in Uganda is 1 hour ahead of Rwanda.
a) If you leave Kabale at 10:00, what will be the local time when
you arrive in Kigali?
b) If you leave Kigali at 17:45, what will be the local time when you
arrive in Kabale?
4. Jean earns UGX 4,000 per hour on weekdays, UGX 4,500 per hour on
Saturdays and UGX 6,000 per hour on Sundays.
The table below lists the hours she worked on each day for one week:
Day No. hours worked
Monday 4
Tuesday 2
Wednesday 8
Thursday 10
Friday 3
Saturday 5
Sunday 2
How much money did Jean earn that week?
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MATHEMATICS
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Situation of Integration: A primary school has two sections, that is,
lower primary (P1-P4) and upper primary (P5-P7). The head teacher of
primary school needs to draw a timetable for both sections. The
sections should start and end their morning lessons at the same time
before break time, start and end their break time at the same time. The
after break lessons should start at the same time. The lunchtime for
both sections should start at the same time.
Support: The time to start lessons for the two sections is 8.00am. The
duration of the lesson for the lower section is 30 minutes and that of
the upper section is 40 minutes.
Resources: Knowledge of fractions, percentages, natural numbers,
factors, multiples, lowest common multiples and of time.
Task: Help the head teacher by drawing the timetable up to lunch
break for the two sections. How many lessons does each section have
up to lunch break?
Express the total number of lessons for the lower primary as a fraction
of the total number of lessons for the whole School. (Consider lessons
up to lunch break)